1. Field of the Invention
The present invention relates to measurement of conductivity (or resistivity) of a solution using a conductivity cell with conductive electrodes.
2. Description of the Prior Art
The measurement of solution conductivity or resistivity is used in a wide variety of applications throughout the process control industry. In particular, the conductivity or resistivity of water may be monitored as an indicator of purity. Absolutely pure water measures approximately 18.25 megohm-centimeters (Meg-cm) resistivity at 25.0.degree. C. Ionic contaminants, most commonly salts, reduce the resistivity of a water sample below this theoretical maximum. A water sample sufficiently free from ionic contaminants that its resistivity is 5 megohm-centimeters or greater is commonly referred to as "high purity" or "ultra pure" water.
Certain industries prefer to measure water purity in terms of conductivity, the mathematical inverse of resistivity. Conductivity is commonly measured in terms of micro mhos/centimeter (.mu.mho/cm) or micro Siemens/centimeter (.mu.S/cm) the units being equivalent, ultra pure water may be described as having a conductivity of 0.2 micro Siemens/centimeter or less.
Conductivity or resistivity of aqueous solutions is typically measured by immersing two conductive surfaces, held in fixed relation to each other, into the solution in question. An electric current is made to flow between the surfaces, through the solution. Assuming the surfaces are themselves perfect conductors, are in perfect electrical contact with the solution, and that the electrical current does not effect the nature of the solution, the solution's conductivity may be calculated as: ##EQU1## Where: Ic=Electric current flowing between the conductive surfaces.
Vc=Voltage across the surfaces. PA0 K="Cell" Constant describing the size and separation of the conductive surfaces.
For flat parallel surfaces of area A and separation L with conductive solution only between the plates, K=L/A. Hence the units megohm-centimeters for resistivity, micro mhos/centimeter for conductivity. The geometrically fixed conductive surfaces, or electrodes are collectively referred to as a conductivity cell.
This basic form of a conductivity cell immersed in a liquid may be electrically modeled as a simple resistor R.sub.C with value equal to the product of solution resistivity .rho. and cell constant K.
For reasons of convenience, modern conductivity cells generally consist of two concentric cylindrical electrodes. An insulating material supports the electrodes and maintains their fixed geometry. Often a temperature monitoring device (RTD or thermistor) is placed in contact with one of the electrodes for purposes of monitoring the temperature of the solution in which the cell is immersed.
A wide variety of conductive materials, ranging from graphite to various types of corrosion resistant steels, are used to make up conductivity cell electrodes. Similarly, a wide variety of materials including glass and epoxy, are used to make up those insulating parts of the cell which hold the electrode geometry fixed. Various minor variations on the concentric cylinder geometry and even some parallel plate electrode configurations are used to ensure a reliable circulation of sample solution in the cell.
All conductivity cells involving electrodes which contact the sample solution, regardless of geometry or material, share the same basic model and the same shortcomings and limitations.
The simple conductivity cell model described above is not adequate for the majority of real-world conductivity measurement applications, particularly where high purity water measurement is involved.
Several problems limit the accuracy of the simple model, the most significant of these being polarization. Polarization is the result of chemical activity between the electrode and solution in the presence of the electrical current used to make the conductivity measurement. A voltage potential, Vcell, must be added to the simple cell model (in series with the resistance) in order to account for polarization effects.
The crux of the polarization problem is that it is nearly impossible to characterize for any real-world application. The magnitude of Vcell is a function of a number of variables including: the electrical current flowing through the cell, temperature, time, electrode material, and solution chemistry. The last of these variables is always an unknown in practical applications, or it would be unnecessary to make the conductivity measurement in the first place.
In short, the error introduced by polarization in such simple conductivity measurement schemes is also virtually impossible to characterize. For this reason, all but the most crude of conductivity measurement schemes employ a somewhat more sophisticated measurement technique.
Virtually all modern conductivity measurement systems now operate their conductivity cells with an alternating current drive signal, in place of a simple direct current drive. Conductivity cells are driven at frequencies ranging from 50 to 60 Hz up to several kHz and at low current levels, typically under 100 micro-amperes. Since the error voltage due to polarization is a direct function both of electrical current magnitude and of the length of time during which current is applied to the conductivity cell, a measurement technique using an alternating current electrical drive of limited magnitude does indeed serve to reduce the error due to polarization.
Although it limits error due to polarization, an alternating current conductivity measurement scheme brings with it new sources of error. Specifically, the capacitive components of the conductivity cell must now be considered. In particular, the inter-lead capacitance C.sub.W for connecting the cell to driver electronics, the electrode-solution interface capacitance C.sub.E, the cell interelectrode capacitance C.sub.C, and the electrode-solution interface resistance R.sub.E all have an effect on the cell voltage.
Typical values for a common 0.01 cell constant conductivity cell according to the circuit in FIG. 1 in a high purity water application are as follows: ##EQU2## Empirical studies indicate that for high purity water, C.sub.E is not a constant, but is related to the resistivity of the measured solution, and the available surface area of the cell electrodes.
For example, a currently available "glass" conductivity cell with constant K=0.01 and platinum electrodes plated with highly porous platinum black, provides C.sub.E &gt;1 microfarad in solution with .rho.=18.3 megohm-cm. Another electrode employing titanium-palladium electrodes, popular with industrial users for its durability, provides C.sub.E .perspectiveto.0.1 microfarad in the same solution.
The effects of all resistance and capacitance components of the conductivity cell model must be considered in order to obtain the maximum accuracy in the measurement of solution conductivity or resistivity.
The majority of conductivity measurement systems currently on the market do not address the conductivity cell error terms C.sub.W, C.sub.C, C.sub.E and R.sub.E. Rather, the conductivity cell is treated as the simple model (i.e. only resistance R.sub.C) and the neglected error terms are allowed to remain present in the system conductivity or resistivity output display.
In general, the error induced by C.sub.W, the cable inter-lead capacitance, may be limited by limiting the length of cable recommended for use in connecting the conductivity cell to the system drive and measurement electronics. In addition, some systems provide a gain adjustment as part of the measurement electronics. Such an adjustment may be used to calibrate the conductivity measurement system at one point, but does little to improve system accuracy over a broad range of conductivity or resistivity values.
The most sophisticated of presently available conductivity measurement systems employ a "center-sampling" technique to reduce the effects of the previously described error terms. To employ this technique, the conductivity cell is driven with a square-wave signal in the 50 to 1,000 Hz range. Alternatively, a trapezoidally shaped drive waveform may be used with similar results.
Cell voltage waveform is the resultant voltage across the conductivity cell when the square wave drive signal is placed across the cell through a reference resistor R.sub.REF. If the conductivity cell were a purely resistive device, the conductivity cell voltage would be a simple square wave with magnitude, V0: ##EQU3## Because of the effects of the cell capacitance terms, most significantly C.sub.W and C.sub.E, the cell output voltage waveform is distorted. The effects of C.sub.W are significant during the early part of the waveform where it causes a non-zero risetime in the conductivity cell output voltage waveform.
The effects of C.sub.W may be essentially eliminated by (a) limited cable length such that C.sub.W 's time constant is much less than .tau./8 (where .tau. is the period of one cycle of the drive waveform), and (b) "center-sampling" the cell output voltage waveform. Center-sampling is achieved by generating a logic signal which is active only between times .tau./8 and 3.tau./8, the center of the positive half-waveform. The sampled voltage is defined only during the sampling interval, .tau./8.ltoreq.time.ltoreq.3.tau./8.
Since C.sub.W 's time constant is kept short, its effects are negligible by time .tau./8 when the sampling interval begins. By averaging the cell voltage output only between times .tau./8 and 3.tau.8, the effects of C.sub.W are kept to negligible levels.
Center-sampling also reduces, but does not eliminate, the effects of C.sub.E on the measured cell output. For most applications the drive frequency is chosen such that: ##EQU4## and the effects of R.sub.E can be ignored.
From the model it is derived that: ##EQU5## Where: V.sub.0 =Average cell output voltage sampled between .tau./8 and 3.tau./8.
R.sub.REF =Reference resistor PA1 R.sub.C =K.times..rho. PA1 V.sub.IM =Magnitude of the square wave drive signal
If the effects of C.sub.E are neglected (C.sub.E assumed to be infinitely large), the error term disappears, the model reduces to that of the simple cell, and the solution resistivity may be calculated from the measured quantity, V.sub.0, and the known values for K, R.sub.R, and V.sub.IM.
This technique Works quite well assuming the effect of C.sub.E, as reduced by the center-sampling technique, actually is negligible.
This is generally the case for cells using platinum electrodes plated with platinum-black. Such cells, because of their high electrode surface porosity, exhibit large C.sub.E values. This reduces the error term to a level which is, for most cases, insignificant.
Platinum/platinum-black electrodes, however, are too fragile to be suitable for most industrial applications. Cells using electrodes made of tough alloys, such as titanium-palladium, are favored by industrial users for their resistance to both physical and chemical attack.
This type of electrode is much less porous, and exhibits a much lower C.sub.E value than a comparable platinum/platinum-black electrode. In addition, as the cell is used, the electrodes often become covered with precipitate from the measured solution. This tends to reduce the value of C.sub.E, and hence increases the error term of Equation 5 still further.
In actual use, conductivity cells using titanium-palladium electrodes may introduce errors of up to one percent into the resistivity or conductivity measurement system when new. This error value may increase undetected to five percent or more over the life of the conductivity cell, as precipitate gradually forms on the cell electrodes. A five percent error is undesirable in most applications and unacceptable in many.